Direct methods in variational field theory

We show that the Weierstrass–Hilbert classical field theory can be strengthened. Namely, for each extremal field, it is true that if an extremal is an element of the field then a minimum is attained in the class of Sobolev functions with the same boundary data as for the extremal and with graphs in the set covered by the field. This result remains valid if one of the extremals is singular. If there is a field containing more than one singular extremal then each of these extremals defines the minimization problem having no solution in the class of Lipschitz functions with graphs in the set covered by the field.